Find $\frac{\partial ^ 2 z}{\partial x^2}$ of $$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ Clearily I can see that:
$$\frac{2x}{a^2}dx + \frac{2y}{b^2}dy + \frac{2z}{c^2}dz = 0$$
Which makes :
$$\frac{\partial z }{\partial x} = -\frac{c^2x}{a^2z}$$
Differentiating with respect to $x$ again we get:
$$\frac{\partial ^2z }{\partial x^2} = -\frac{c^2}{a^2z}$$
However apparently the answer is $$\frac{\partial ^2z }{\partial x^2} = = -\frac{c^4(b^2-y^2)}{a^2b^2z^3}$$
What is the flaw in logic?
It was fine until you transitioned to the second derivative. Using the quotient rule, $$\frac{\partial ^2z }{\partial x^2} = -\frac{c^2(z-x\frac{\partial z }{\partial x})}{a^2z^2}$$ Now just substitute the first partial derivative and simplify. Also, your first step is not really necessary.